CN112344795A - Terminal guidance method for predetermined time convergence - Google Patents

Terminal guidance method for predetermined time convergence Download PDF

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CN112344795A
CN112344795A CN202011235100.4A CN202011235100A CN112344795A CN 112344795 A CN112344795 A CN 112344795A CN 202011235100 A CN202011235100 A CN 202011235100A CN 112344795 A CN112344795 A CN 112344795A
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方洋旺
邓天博
王志凯
吴自豪
马文卉
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Abstract

The invention discloses a terminal guidance method for predetermined time convergence, which is characterized in that a time function with infinitely increased terminal time is designed to zoom a system state, and then a controller capable of stabilizing the system in the zooming state is designed, so that the original state is adjusted within a predetermined limited time. Compared with the prior fixed time method, the method does not depend on the initial condition any more, and can realize adjustment within the preset limited time even if non-zero uncertain nonlinearity exists; the method can be used as a basic guidance law of the cooperative guidance law and lays a foundation for the design of the advanced cooperative guidance law.

Description

Terminal guidance method for predetermined time convergence
Technical Field
The invention belongs to the technical field of navigation, and particularly relates to a terminal guidance method.
Background
In recent years, in order to achieve efficient missile striking and some tactical applications, research on guidance laws with terminal entry angle constraints has received much attention. Most of the existing guidance laws can meet the expected entrance angle constraint, the line-of-sight angular velocity can be converged to zero, and the problem of adjusting the convergence velocity is not well solved. In practical engineering, the line-of-sight angular velocity needs to converge to zero before the target is hit, and the rapidity of system convergence needs to be considered, so that a guidance rule with limited time convergence needs to be designed.
At present, some scholars propose a guidance law method with limited time convergence, but the convergence speed of the angular velocity of the visual line in the method is limited by the initial state of the missile; in order to overcome the problem, some scholars propose a fixed-time convergence guidance law, although the convergence speed of the method is irrelevant to the initial state of the missile, the convergence time of the method cannot be adjusted by controlling relevant variables, and the convergence of the line-of-sight angle of the missile to a desired value before the missile hits a target cannot be guaranteed. Therefore, designing an end guidance law with an adjustable convergence time and an entrance angle constraint is of great significance in engineering.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a terminal guidance method for predetermined time convergence, which scales the system state by designing a time function with infinitely increased terminal time, and then designs a controller capable of stabilizing the system in the scaling state, thereby realizing the adjustment of the original state in a predetermined limited time. Compared with the prior fixed time method, the method does not depend on the initial condition any more, and can realize adjustment within the preset limited time even if non-zero uncertain nonlinearity exists; the method can be used as a basic guidance law of the cooperative guidance law and lays a foundation for the design of the advanced cooperative guidance law.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
step 1: establishing a second-order guidance model with an entrance angle constraint;
a model of the motion relationship between the missile and the target in a two-dimensional plane is defined as follows:
Figure BDA0002766557380000011
Figure BDA0002766557380000012
wherein R is the distance between the missile and the target, q is the sum
Figure BDA0002766557380000013
Respectively, the angle of the line of sight of the bullet eye and the rate of the angle of sight of the bullet eye, VMAnd VTThe velocity of the missile and the target, thetaMAnd thetaTTrajectory inclination angles of the missile and the target, respectively;
the derivation of equation (2) and the substitution of equation (1) yields:
Figure BDA0002766557380000021
wherein, aMAnd aTNormal acceleration of the missile and target, respectively;
the state variables defining the guidance model are respectively x1＝q-qfAnd
Figure BDA0002766557380000022
wherein q isfA desired entrance angle; the state space equation of relative movement of the bullets in the longitudinal plane is obtained by the formula (3):
Figure BDA0002766557380000023
assuming that the speed of the target is unchanged; considering u as the control input to the system, equation (4) is written as:
Figure BDA0002766557380000024
wherein the content of the first and second substances,
Figure BDA0002766557380000025
in order to control the amount of the liquid,
Figure BDA0002766557380000026
an acceleration compensation term for the target maneuver;
assuming that f (x) satisfies:
|f(x)|≤d(t)ψ(x) (6)
wherein the content of the first and second substances,
Figure BDA0002766557380000027
a perturbation whose boundary is unknown; psi (x) is a positive scalar value continuous function;
step 2: designing a mu function and constructing an auxiliary state equation;
construct the following monotonically increasing function over [0, T):
Figure BDA0002766557380000028
in the formula: m and n are positive integers, T is a time variable, T is a desired convergence time,
Figure BDA0002766557380000029
if a continuous differentiable function V: [0, t) → [0, + ∞) satisfies:
Figure BDA00027665573800000210
in the formula: k. λ is a positive real number; then
Figure BDA0002766557380000031
Where ζ is a monotonically decreasing function:
Figure BDA0002766557380000032
and has the following properties: ζ (0) ═ 1, ζ (T) ═ 0;
defining the auxiliary state variable w using the scaling function μ of equation (7)1(t),w2(t) and obtaining the following equation of state: :
Figure BDA0002766557380000033
and step 3: designing a final guidance law with predetermined time convergence;
the constructor z:
z＝w2(t)+k1w1(t) (12)
wherein k is1Is a positive real number;
from equations (5), (6) and (12), a guidance law with predetermined time convergence is obtained:
Figure BDA0002766557380000034
where θ is a positive real number.
Preferably, m is 2 and n is 2.
Due to the adoption of the terminal guidance method with the predetermined time convergence, the following beneficial effects are achieved:
1. compared with the prior fixed time method, the method of the invention does not depend on the initial condition any more, and can realize the adjustment within the preset limited time even if the non-zero uncertain nonlinearity exists.
2. The bending degree of the trajectory can be changed by changing the value of the preset time T in the guidance law designed by the method, so that the attack time of the missile is influenced, and therefore the terminal guidance law designed by the method can be used as a basic guidance law of the cooperative guidance law and lays a foundation for the design of the advanced cooperative guidance law.
Drawings
FIG. 1 is a schematic diagram showing the relative motion relationship between the missile and the target according to the invention.
FIG. 2 is a schematic diagram of the trajectory of the missile of the present invention.
FIG. 3 is a schematic diagram showing the variation curve of the distance of the projectile during the movement of the missile of the invention.
FIG. 4 is a schematic view of the change of the angle of view of the missile of the invention during its motion.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
As shown in fig. 1, an end-guidance method for predetermined time convergence includes the following steps:
step 1: establishing a second-order guidance model with an entrance angle constraint;
a model of the motion relationship between the missile and the target in a two-dimensional plane is defined as follows:
Figure BDA0002766557380000041
Figure BDA0002766557380000042
wherein R is the distance between the missile and the target, q is the sum
Figure BDA0002766557380000043
Respectively, the angle of the line of sight of the bullet eye and the rate of the angle of sight of the bullet eye, VMAnd VTThe velocity of the missile and the target, thetaMAnd thetaTTrajectory inclination angles of the missile and the target, respectively;
the derivation of equation (2) and the substitution of equation (1) yields:
Figure BDA0002766557380000044
wherein, aMAnd aTNormal acceleration of the missile and target, respectively;
defining guidance modelsThe state variables are each x1＝q-qfAnd
Figure BDA0002766557380000045
wherein q isfA desired entrance angle; the state space equation of relative movement of the bullets in the longitudinal plane is obtained by the formula (3):
Figure BDA0002766557380000046
assuming that the speed of the target is unchanged, maneuvering avoidance is realized by changing the speed direction, and the normal acceleration a of the targetTCan be estimated by means of an observer; regarding u as a control input of the system in the design of the guidance law, writing equation (4) as:
Figure BDA0002766557380000047
wherein the content of the first and second substances,
Figure BDA0002766557380000048
in order to control the amount of the liquid,
Figure BDA0002766557380000049
an acceleration compensation term for the target maneuver;
assuming that f (x) satisfies: l f (x) l < d (t) ψ (x) (6)
Wherein the content of the first and second substances,
Figure BDA0002766557380000051
a perturbation whose boundary is unknown; psi (x) is a positive scalar value continuous function;
step 2: designing a mu function and constructing an auxiliary state equation;
construct the following monotonically increasing function over [0, T):
Figure BDA0002766557380000052
in the formula: m and n are positive integers,
Figure BDA0002766557380000053
the guidance model applied in the invention is of second order, so that m is 2, and n is 2;
if a continuous differentiable function V: [0, t) → [0, + ∞) satisfies:
Figure BDA0002766557380000054
in the formula: k. λ is a positive real number; then
Figure BDA0002766557380000055
Where ζ is a monotonically decreasing function:
Figure BDA0002766557380000056
and has the following properties: ζ (0) ═ 1, ζ (T) ═ 0;
the above process is derived from:
by integrating the left and right sides of the differential inequality (8):
Figure BDA0002766557380000057
the second term on the right side of the upper equation yields:
Figure BDA0002766557380000061
by
Figure BDA0002766557380000062
And combining the formula to finally obtain:
Figure BDA0002766557380000063
in order to realize timing adjustment of the second-order guidance system, the second-order guidance state equation is transformed, and the following auxiliary state equation is obtained by using the scaling function mu of the formula (7):
Figure BDA0002766557380000064
the function μ used in the present invention is characterized in that when T → T, the function μ1Approaching infinity from 1 due to x1(t) is gradually approaching zero, so w1(t)＝μ1(t)x1(t) is convergent.
And step 3: obtaining a terminal guidance law with preset time convergence;
considering that the model in the invention is a second-order system, the state equation is shown in the formula (5), the state equation of the system is converted into the state equation containing the mu function in the formula (11) by introducing the mu function, and the preset time convergence controller is designed based on the new state equation. The purpose of the finite time convergence control law designed by the invention is to converge the line-of-sight angle and the line-of-sight angular rate within a preset time T to an arbitrary small interval. However, the line-of-sight angular rate converges to zero, which does not mean the end of the guidance process, and the guidance process continues when the distance between the missile and the target is not zero within the time T, but the control performance of the line-of-sight angle constraint is achieved at the moment, and when T > T, the missile is finely adjusted by using proportional guidance, so that the missile can hit the target according to the preset performance.
The constructor z:
z＝w2(t)+k1w1(t) (12)
wherein k is1Is a positive real number;
from equations (5), (6) and (12), a guidance law with predetermined time convergence is obtained:
Figure BDA0002766557380000071
where θ is a positive real number.
For the second-order guidance model shown in the formula (5) and the auxiliary state equation shown in the formula (11), the controller shown in the formula (13) is designed to realize that the line-of-sight angle of the missile is converged at the preset time T, and the target is hit at the preset incidence angle:
Figure BDA0002766557380000072
wherein the content of the first and second substances,
Figure BDA0002766557380000073
k＝0.1；θ＝0.1；
Figure BDA0002766557380000074
λ ═ 1; n is 3; t-12/16/20; z is an auxiliary variable as shown in formula (12), wherein k1＝0.05；w1＝μ1x1Wherein x is1＝q-qf，qf＝-20°。
A control parameter k >0 and satisfies
ρk>γ1/4
Where ρ is a constant greater than 0.
γ1Satisfies the following conditions:
Figure BDA0002766557380000075
Figure BDA0002766557380000076
they are all normal numbers.
The control parameter theta is more than or equal to theta*，θ*Can be expressed as:
Figure BDA0002766557380000077
wherein v satisfies x (t) v (t) w (t);
Figure BDA0002766557380000081
Figure BDA0002766557380000082
J1＝[1,0]；J2＝[0,1]；e2＝[0,1]T。
the above procedure can be derived as follows:
selecting the Lyapunov function V ═ z2And/2, obtaining a derivative:
Figure BDA0002766557380000083
wherein the content of the first and second substances,
Figure BDA0002766557380000084
using the young inequality, one can obtain:
Figure BDA0002766557380000085
due to the fact that
Figure BDA0002766557380000086
Then
Figure BDA0002766557380000087
And due to
Figure BDA0002766557380000088
From equation (17) and the young's inequality, it is possible to obtain:
Figure BDA0002766557380000089
where ρ > 0.
Figure BDA00027665573800000810
Due to k1>0 and according to equation (18),
Figure BDA00027665573800000811
substituting formula (18) for formula (17) yields:
Figure BDA00027665573800000812
according to formulae (15), (16) and (19), we obtain:
Figure BDA0002766557380000091
finally, from the formula (7) and the formula (9), it is found that when T ∈ [0, T ], there is
Figure BDA0002766557380000092
According to the relationship between the auxiliary variable z (T) and the state variable, x (T) can be obtained to converge to an arbitrary small interval at the preset time T.
The specific embodiment is as follows:
the method verifies the effectiveness of the designed algorithm by attacking a maneuvering target with a missile, and under the action of the designed control law, the line-of-sight angle of the missile converges to a preset entering angle within a specified limited time, and the line-of-sight angular rate converges to zero within a specified time.
And (4) designing a simulation experiment according to the guidance law designed in the step (3), and carrying out simulation analysis. Wherein the initial position of the missile is (-5000,3800) m, and the speed of the missile is Vm0400m/s, and the initial lead angle of the missile is 40 degrees; the initial position of the target is (0,0) m, and the target velocity Vt0100m/s, the initial lead angle of the target is-10 degrees, the target is driven by sine wave, and the normal acceleration is aT2 g cos (0.5 t), where g is the gravitational acceleration constant, g is 9.8m/s2And t is the time of flight of the target.
The simulation results are shown in fig. 2 to 4: as can be seen in FIG. 2, the missiles have different trajectories under different T, and the larger the T, the more the initial trajectory of the missile is bent, and the longer the attack time is; but eventually the missile can hit the target accurately. Fig. 3 shows the relative distance change between the missile and the target under different T, and it can be seen that the missile can accurately hit the target in all three cases, but the time required for the missile to hit the target is shorter as T is increased. Fig. 4 shows the change of the line-of-sight angle of the missile under the action of different T, and it can be seen that the line-of-sight angle of the missile can be converged to a desired value at any time T.

Claims (2)

1. An end-guidance method for predetermined time convergence, comprising the steps of:
step 1: establishing a second-order guidance model with an entrance angle constraint;
a model of the motion relationship between the missile and the target in a two-dimensional plane is defined as follows:
Figure FDA0002766557370000011
Figure FDA0002766557370000012
wherein R is the distance between the missile and the target, q is the sum
Figure FDA0002766557370000013
Respectively, the angle of the line of sight of the bullet eye and the rate of the angle of sight of the bullet eye, VMAnd VTThe velocity of the missile and the target, thetaMAnd thetaTTrajectory inclination angles of the missile and the target, respectively;
the derivation of equation (2) and the substitution of equation (1) yields:
Figure FDA0002766557370000014
wherein, aMAnd aTNormal acceleration of the missile and target, respectively;
the state variables defining the guidance model are respectively x1＝q-qfAnd
Figure FDA0002766557370000015
wherein q isfA desired entrance angle; the state space equation of relative movement of the bullets in the longitudinal plane is obtained by the formula (3):
Figure FDA0002766557370000016
assuming that the speed of the target is unchanged; considering u as the control input to the system, equation (4) is written as:
Figure FDA0002766557370000017
wherein the content of the first and second substances,
Figure FDA0002766557370000018
in order to control the amount of the liquid,
Figure FDA0002766557370000019
an acceleration compensation term for the target maneuver;
assuming that f (x) satisfies:
|f(x)|≤d(t)ψ(x) (6)
wherein the content of the first and second substances,
Figure FDA00027665573700000110
a perturbation whose boundary is unknown; psi (x) is a positive scalar value continuous function;
step 2: designing a mu function and constructing an auxiliary state equation;
construct the following monotonically increasing function over [0, T):
Figure FDA0002766557370000021
in the formula: m and n are positive integers, T is a time variable, T is a desired convergence time,
Figure FDA0002766557370000022
if a continuous differentiable function V: [0, t) → [0, + ∞) satisfies:
Figure FDA0002766557370000023
in the formula: k. λ is a positive real number; then
Figure FDA0002766557370000024
Where ζ is a monotonically decreasing function:
Figure FDA0002766557370000025
and has the following properties: ζ (0) ═ 1, ζ (T) ═ 0;
defining the auxiliary state variable w using the scaling function μ of equation (7)1(t),w2(t) and obtaining the following equation of state: :
Figure FDA0002766557370000026
and step 3: designing a final guidance law with predetermined time convergence;
the constructor z:
z＝w2(t)+k1w1(t) (12)
wherein k is1Is a positive real number;
from equations (5), (6) and (12), a guidance law with predetermined time convergence is obtained:
Figure FDA0002766557370000027
where θ is a positive real number.
2. The end guidance method for predetermined time convergence according to claim 1, wherein m is 2 and n is 2.
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